• Nielsen, M. A., Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).


    Google Scholar
     

  • Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).

    Article 
    ADS 

    Google Scholar
     

  • Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Herman, D. et al. Quantum computing for finance. Nat. Rev. Phys. 5, 450–465 (2023).

    Article 

    Google Scholar
     

  • Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Das Sarma, S., Deng, D.-L. & Duan, L.-M. Machine learning meets quantum physics. Phys. Today 72, 48–54 (2019).

    Article 

    Google Scholar
     

  • Gottesman, D. E. Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology (1997).

  • Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Flühmann, C. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).

    Article 
    ADS 

    Google Scholar
     

  • Egan, L. et al. Fault-tolerant control of an error-corrected qubit. Nature 598, 281–286 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Ryan-Anderson, C. et al. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X 11, 041058 (2021).


    Google Scholar
     

  • de Neeve, B., Nguyen, T.-L., Behrle, T. & Home, J. P. Error correction of a logical grid state qubit by dissipative pumping. Nat. Phys. 18, 296–300 (2022).

    Article 

    Google Scholar
     

  • Paetznick, A. et al. Demonstration of logical qubits and repeated error correction with better-than-physical error rates. Preprint at https://arxiv.org/abs/2404.02280 (2024).

  • Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014).

    Article 
    ADS 

    Google Scholar
     

  • Abobeih, M. H. et al. Fault-tolerant operation of a logical qubit in a diamond quantum processor. Nature 606, 884–889 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Chang, X.-Y. et al. Hybrid entanglement and bit-flip error correction in a scalable quantum network node. Nat. Phys. 22, 583–589 (2025).

    Article 

    Google Scholar
     

  • Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Reichardt, B. W. et al. Logical computation demonstrated with a neutral atom quantum processor. Preprint at https://arxiv.org/abs/2411.11822 (2024).

  • Marques, J. F. et al. Logical-qubit operations in an error-detecting surface code. Nat. Phys. 18, 80–86 (2022).

    Article 

    Google Scholar
     

  • Zhao, Y. et al. Realization of an error-correcting surface code with superconducting qubits. Phys. Rev. Lett. 129, 030501 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Krinner, S. et al. Realizing repeated quantum error correction in a distance-three surface code. Nature 605, 669–674 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Google Quantum AI et al. Suppressing quantum errors by scaling a surface code logical qubit. Nature 614, 676–681 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Ni, Z. et al. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature 616, 56–60 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Sivak, V. V. et al. Real-time quantum error correction beyond break-even. Nature 616, 50–55 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Gupta, R. S. et al. Encoding a magic state with beyond break-even fidelity. Nature 625, 259–263 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Lacroix, N. et al. Scaling and logic in the colour code on a superconducting quantum processor. Nature 645, 614–619 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Caune, L. et al. Demonstrating real-time and low-latency quantum error correction with superconducting qubits. Preprint at https://arxiv.org/abs/2410.05202 (2024).

  • Eickbusch, A. et al. Demonstration of dynamic surface codes. Nat. Phys. 21, 1994–2001 (2025).

    Article 

    Google Scholar
     

  • Google Quantum AI and Collaborators et al. Quantum error correction below the surface code threshold. Nature 638, 920–926 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Bravyi, S. B. & Kitaev, A. Y. Quantum codes on a lattice with boundary. Preprint at https://arxiv.org/abs/quant-ph/9811052 (1998).

  • Breuckmann, N. P. & Eberhardt, J. N. Quantum low-density parity-check codes. PRX Quantum 2, 040101 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Bravyi, S. et al. High-threshold and low-overhead fault-tolerant quantum memory. Nature 627, 778–782 (2024).

    Article 
    ADS 

    Google Scholar
     

  • Tillich, J.-P. & Zémor, G. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength. IEEE Trans. Inf. Theory 60, 1193–1202 (2014).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Panteleev, P. & Kalachev, G. Asymptotically good quantum and locally testable classical LDPC codes. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022) 375–388 (ACM, 2022).

  • Leverrier, A. & Zemor, G. Quantum Tanner codes. In Proc. 63rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2022) 872–883 (IEEE, 2022).

  • Kim, Y. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023).

    Article 
    ADS 

    Google Scholar
     

  • Jin, F. et al. Topological prethermal strong zero modes on superconducting processors. Nature 645, 626–632 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Roffe, J., White, D. R., Burton, S. & Campbell, E. Decoding across the quantum low-density parity-check code landscape. Phys. Rev. Res. 2, 043423 (2020).

    Article 

    Google Scholar
     

  • Panteleev, P. & Kalachev, G. Degenerate quantum LDPC codes with good finite length performance. Quantum 5, 585 (2021).

    Article 

    Google Scholar
     

  • Chen, Z. et al. Fabrication and characterization of aluminum airbridges for superconducting microwave circuits. Appl. Phys. Lett. 104, 052602 (2014).

    Article 
    ADS 

    Google Scholar
     

  • Song, C. et al. Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits. Science 365, 574–577 (2019).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Google Quantum AI et al. Exponential suppression of bit or phase errors with cyclic error correction. Nature 595, 383–387 (2021).

    Article 

    Google Scholar
     

  • Jeffrey, E. et al. Fast accurate state measurement with superconducting qubits. Phys. Rev. Lett. 112, 190504 (2014).

    Article 
    ADS 

    Google Scholar
     

  • Heinsoo, J. et al. Rapid high-fidelity multiplexed readout of superconducting qubits. Phys. Rev. Appl. 10, 034040 (2018).

    Article 
    ADS 

    Google Scholar
     

  • Sunada, Y. et al. Fast readout and reset of a superconducting qubit coupled to a resonator with an intrinsic Purcell filter. Phys. Rev. Appl. 17, 044016 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Spring, P. A. et al. Fast multiplexed superconducting qubit readout with intrinsic Purcell filtering using a multiconductor transmission line. PRX Quantum 6, 020345 (2025).

    Article 
    ADS 

    Google Scholar
     

  • Sank, D. et al. System characterization of dispersive readout in superconducting qubits. Phys. Rev. Appl. 23, 024055 (2025).

    Article 
    ADS 

    Google Scholar
     

  • McEwen, M. et al. Removing leakage-induced correlated errors in superconducting quantum error correction. Nat. Commun. 12, 1761 (2021).

    Article 
    ADS 

    Google Scholar
     

  • Miao, K. C. et al. Overcoming leakage in quantum error correction. Nat. Phys. 19, 1780–1786 (2023).

    Article 

    Google Scholar
     

  • Bluvstein, D. et al. A fault-tolerant neutral-atom architecture for universal quantum computation. Nature 649, 39–46 (2026).

    Article 

    Google Scholar
     

  • Wang, K. et al. Demonstration of low-overhead quantum error correction codes. Zenodo https://doi.org/10.5281/zenodo.17706106 (2025).